The Daily Telegraph has published a recording of a phone conversation between the Culture Secretary's adviser and one of its reporters investigating her expense claims in 2012, in which the aide "flagged up" her boss' role in the future of press regulation.
The paper has previously reported details of their conversation.
The Sun could be set to follow the lead of other UK newspapers in charging readers to view online content.
Fellow News International publications The Times and Sunday Times introduced a paywall in July 2010.
The Daily Telegraph has said the cross-party deal on press regulation "deserves careful consideration."
It also says that the Prime Minister should admit the plans do involve legislation:
The Culture Secretary receives the backing of the Prime Minister but faces a potentially lengthy investigation into her expenses, reports ITV News' Political Correspondent Alex Forrest:
The Daily Telegraph has published a timeline of the sequence of events over its investigation and subsequent reporting of the Culture Secretary's expense claims "to set the record straight."
It says it was contacted by a source about the story nearly a month ago - on November 17.
It has also published emails between Maria Miller and the paper's editor Tony Gallagher:
The Prime Minister told reporters that Maria Miller has "excellent answers" to all questions being asked about her expenses.
David Cameron has given his backing to Culture Secretary Maria Miller amid questions over her expenses.
"Maria Miller does an absolutely excellent job and she has my full support," the Prime Minister told reporters as he arrived at the EU Summit in Brussels.
"A newspaper has asked her a number of questions and as far as I can see she has excellent answers to all those questions," he added.
Maria Miller has denied that she had used her position overseeing the post-Leveson reforms of press regulation to ward off the Telegraph. She told the Evening Standard:
Mrs Miller's special adviser, Joanna Hindley, reportedly told Telegraph reporters investigating her expenses: "Maria has obviously been having quite a lot of editors' meetings around Leveson at the moment. I am just going to flag up that connection for you to think about."
Maria Miller has said one of the two audits into her expenses had been carried out by former civil servant Sir Thomas Legg - who was called in to review all MPs' claims at the height of the expenses scandal - and the other by the Conservative Party.
Mrs Miller claimed second home allowances of £90,718 - almost the maximum permitted - between 2005 and 2009 towards mortgage payments, bills and other costs relating to a house where her parents had apparently been living since 1996.
Asked by the Evening Standard whether Sir Thomas was aware that her parents were living at her designated second home, Mrs Miller said: "I obviously spoke to the Fees Office about my claims and they were happy that everything was in order."